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More Intensional Versions of Rice’s Theorem

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Computing with Foresight and Industry (CiE 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11558))

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Abstract

Classic results in computability theory concern extensional results: the behaviour of partial recursive functions rather than the programs computing them. We prove a generalisation of Rice’s Theorem concerning equivalence classes of programs and show how it can be used to study intensional properties such as time and space complexity. While many results that follow from our general theorems can - and have - been proved by more involved, specialised methods, our results are sufficiently simple that little work is needed to apply them.

A very preliminary version of this paper first appeared at the DICE ’16 workshop [12].

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Notes

  1. 1.

    For the technical development, we need only assume \(\mathcal {P}\) to be r.e.  but morally we are interested in concrete programming languages where \(\mathcal {P}\) will always be a decidable set.

  2. 2.

    See, e.g., [8] for an easily readable account.

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Author information

Authors and Affiliations

  1. LIPN, Université Paris 13, Villetaneuse, France

    Jean-Yves Moyen

  2. Department of Computer Science, University of Copenhagen (DIKU), Copenhagen, Denmark

    Jakob Grue Simonsen

Authors
  1. Jean-Yves Moyen
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  2. Jakob Grue Simonsen
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Corresponding author

Correspondence to Jakob Grue Simonsen .

Editor information

Editors and Affiliations

  1. Christian Albrechts University of Kiel, Kiel, Germany

    Florin Manea

  2. Durham University, Durham, UK

    Barnaby Martin

  3. Durham University, Durham, UK

    Daniël Paulusma

  4. Università degli Studi di Milano, Milan, Italy

    Giuseppe Primiero

A Material Omitted from the Main Text

A Material Omitted from the Main Text

Proof

(Full proof of intrication from Example 3). We prove that the switching family \(( \pi _{a,b} )_{(a,b) \in (\cup F) \times (\cup G)}\) is \(\textsc {RE} \)-F-G-intricated with \(\Bumpeq _\mathtt {s}\):

If \(\mathtt {p}\in \cup F\), \(\mathtt {q}\in \cup G\) and \(T' \in \mathcal {T}\) such that \(A_{\mathtt {p},\mathtt {q}} \subseteq \pi _{\mathtt {p},\mathtt {q}} ^{-1}(T')\), there are two cases:

  • If \(\phi _{\mathtt {p}} = \phi _{\mathtt {q}}\), then \(A_{\mathtt {s},\mathtt {p}} = B_{\mathtt {s},\mathtt {q}}\), and there is clearly no set separating \(A_{\mathtt {s},\mathtt {p}}\) from \(B_{\mathtt {s},\mathtt {q}}\).

  • If \(\phi _{\mathtt {p}} \ne \phi _{\mathtt {q}}\), there are again two cases:

    • If r does not halt on n, then \(\phi _{s_{r,n}} = \phi _{\mathtt {p}} = \phi _{\mathtt {s}} \in F\), and thus \((r,n) \in A_{\mathtt {p},\mathtt {q}}\). Conversely, if \((r,n) \in A_{\mathtt {p},\mathtt {q}}\), then \(\phi _{s_{r,n}} = \phi _{\mathtt {s}}\), and as \(\lnot (\mathtt {q}\Bumpeq \mathtt {s})\), this implies that r does not halt on n.

    • If r halts on n, then \(\phi _{s_{r,n}} \bumpeq \phi _{\mathtt {q}}\). As \(\mathtt {q}\in \cup G\), we have \(\lnot (\mathtt {q}\Bumpeq \mathtt {s})\), and thus \(\lnot (\phi _{s_{r,n}} \Bumpeq \mathtt {s})\), whence in particular \(\phi _{s_{r,n}} \ne \phi _{\mathtt {s}}\), and hence \(\phi _{s_{r,n}} \Bumpeq _\mathtt {s}\mathtt {q}\). Thus, \((r,n) \in B_{\mathtt {p},\mathtt {q}}\). Conversely, if \((r,n) \in B_{\mathtt {p}, \mathtt {q}}\), \(s_{r,n} \Bumpeq _\mathtt {s}\mathtt {q}\) which–as \(\lnot (\mathtt {q}\Bumpeq \mathtt {p})\)–implies \(\phi _{s_{r,n}} \ne \phi _{\mathtt {s}}\) and hence that r halts on n.

    Thus \((r,n) \in A_{\mathtt {p},\mathtt {q}}\) iff r does not halt on n, and \((r,n) \in B_{\mathtt {p}, \mathtt {q}}\) iff r halts on n. Hence, \(A_{\mathtt {p},\mathtt {q}} \cup B_{\mathtt {p},\mathtt {q}} = \mathcal {P}\times \mathcal {N}\), and \(A_{\mathtt {p},\mathtt {s}}\) are \(B_{\mathtt {p},\mathtt {s}}\) is co-r.e.-complete and r.e.-complete, respectively. If there were an an r.e. set C such that \(A_{\mathtt {p},\mathtt {q}} \subseteq C\) and \(B_{a,b} \cap C = \emptyset \), then \(A_{\mathtt {p},\mathtt {q}} \cup B_{\mathtt {p},\mathtt {q}} = \mathcal {P}\times \mathcal {N}\) implies \(A_{a,b} = C\), whence \(A_{a,b }\) is r.e., contradicting that it is co-r.e.-complete.    \(\square \)

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Moyen, JY., Simonsen, J.G. (2019). More Intensional Versions of Rice’s Theorem. In: Manea, F., Martin, B., Paulusma, D., Primiero, G. (eds) Computing with Foresight and Industry. CiE 2019. Lecture Notes in Computer Science(), vol 11558. Springer, Cham. https://doi.org/10.1007/978-3-030-22996-2_19

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  • DOI: https://doi.org/10.1007/978-3-030-22996-2_19

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